Mini Tutorial MT-220

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Rev. 0 | Page 1 of 3

Multiple Feedback Filters by Hank Zumbahlen, Analog Devices, Inc.

IN THIS MINI TUTORIAL Three sample multiple feedback filters are designed in this

mini tutorial, one in a series of mini tutorials describing

discrete circuits for precision op amps.

The multiple feedback filter, a popular configuration, uses an op amp as an integrator as shown in Figure 1. Therefore, the dependence of the transfer function on the op amp parameters is greater than in the Sallen-Key realization.

It is difficult to generate high Q, high frequency sections due to the limitations of the open-loop gain of the op amp. A rule of thumb is that the open-loop gain of the op amp should be at least 20 dB (10) above the amplitude response at the resonant (or cut-off) frequency, including the peaking caused by the Q of the filter. The peaking due to Q causes an amplitude, A0

A0 = H Q (1)

where H is the gain of the circuit. The multiple feedback filter inverts the phase of the signal. This is equivalent to adding the resulting 180 phase shift to the phase shift of the filter itself.

Figure 1. Multiple Feedback Low-Pass Filter

The maximum-to-minimum component value ratio is higher in the multiple feedback case than in the Sallen-Key realization. The design equations for the multiple feedback low-pass filter are given in the Multiple Feedback Low-Pass Design Equations section.

Comments regarding the multiple feedback low-pass filter can apply to the high-pass filter as well (see Figure 2). Again, resistors and capacitors are swapped to convert the low-pass case to the high-pass case. The design equations for the multiple feedback high-pass filter are given in the Multiple Feedback High-Pass Design Equations section.

Figure 2. Multiple Feedback High-Pass Filter

The design equations for the multiple feedback band-pass filter are given in the Multiple Feedback Band-Pass Design Equations section

Figure 3. Multiple Feedback Band-Pass Filter

This circuit is widely used in low Q (< 20) applications. It allows some tuning of the resonant frequency, F0, by making R2 variable. Q can be adjusted (with R5) as well, but this also changes F0.

One way to tune the filter F0 is by monitoring the output of the filter with the horizontal channel of an oscilloscope, with the input to the filter connected to the vertical channel. The display will be a Lissajous pattern. This pattern is an ellipse that collapses to a straight line at resonance, since the phase shift is 180. In addition, adjust the output for maximum output, which occurs at resonance; however, this is usually not as precise, especially at lower values of Q where there is a less pronounced peak.

MULTIPLE FEEDBACK LOW-PASS DESIGN EQUATIONS

200

20 2

++

ssH

Figure 4.

R1

R4

C2

R3

C5

OUT

IN

1042

5-00

1

C1

R5

C3

C4

OUT

IN

R2

1042

5-00

2

R1

R5

C3

C4

OUT

IN

R2

1042

5-00

3

R1

R4

C2

R3

C5

OUT

IN

1042

5-00

4

MT-220 Mini Tutorial

Rev. 0 | Page 2 of 3

52431

41

31

11

21

52311

2

CCRRRRRCss

CCRRH

VV

IN

O

+

+++

=

To design the filter, choose C3.

Then

k = 2 F0 C5

( ) 512

42 CHC +=

kHR

21 =

( ) kHR 123 +=

kR

24 =

MULTIPLE FEEDBACK HIGH-PASS DESIGN EQUATIONS

200

2

2

++

sssH

Figure 5.

43521

543431

41

2

2

CCRRRCCCCCss

CCs

VV

IN

O

+

+++

=

To design the filter, choose C1.

Then

k = 2 F0 C1

C3 = C1

HCC 14 =

+

=

Hk

R12

2

kH

HR

+

=

125

MULTIPLE FEEDBACK BAND-PASS DESIGN EQUATIONS

200

20

++

sssH

Figure 6.

++

++

=

21

11

4351

54343

411

2

RRCCRRCCCCss

CRs

VV

IN

O

To design the filter, choose C3.

Then

k = 2 F0 C3

C4 = C3

kHR 11=

( ) kHQR = 212

kQR 25 =

REFERENCES Jung, Walter G., editor. 2002. Op Amp Applications Handbook,

Newnes, ISBN 0-916550-26-5.

Kester, Walt, editor. 1992. Amplifier Applications Guide, Analog Devices, Inc. ISBN 0-916550-10-9.

Kester, Walt, editor. 2004. Analog-Digital Conversion, Analog Devices, Inc. ISBN 0-916550-27-3.

Williams, A. B. 1981. Electronic Filter Design Handbook, McGraw-Hill. ISBN 0-07-070430-9.

Zumbahlen, Hank, editor. 2007. Basic Linear Design, Analog Devices, Inc. ISBN 0-916550-28-1.

Zumbahlen, Hank. Phase Relations in Active Filters. Analog Dialogue, Vol. 14, No. 4, 2008.

Zumbahlen, Hank, editor, 2008. Linear Circuit Design Handbook, Newnes, ISBN 978-0-7506-8703-4.

C1

R5

C3

C4

OUT

IN

R2

1042

5-00

5

R1

R5

C3

C4

OUT

IN

R2

1042

5-00

6

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Mini Tutorial MT-220

Rev. 0 | Page 3 of 3

REVISION HISTORY 3/12Revision 0: Initial Version

2012 Analog Devices, Inc. All rights reserved. Trademarks and registered trademarks are the property of their respective owners. MT10425-0-3/12(0)

In This Mini TutorialMultiple Feedback Low-Pass Design EquationsMultiple Feedback High-Pass Design EquationsMultiple Feedback Band-Pass Design EquationsReferences